Ok, Mesopotamia, I'll bite. I am NOT an expert on ancient history but I do have a fairly solid grasp of premodern financial history.

The actually crucial piece for trade mentioned in your question is currency, not coinage. In this particular historical context, "currency" is mostly interchangeable with "money". Currencies functioned as mediums of exchange, units of account and stores of value. Mesopotamians used currencies as well as credit. Indeed, Michael Hudson goes so far as to start his recent book chapter on the ancient origin of money with this: "Neolithic and Bronze Age economies operated mainly on credit".

Take the Laws of Eshnunna, written in Akkadian on two cuneiform tablets probably in the 18th century BC. It states how certain commodities can be purchased with a certain amount of silver, eg. 1 gur of barley for 1 shekel of silver, and 12 sila of oil for 1 shekel of silver. Here the 1 shekel (little over 8g) of silver is a standardized unit of account that allows the contemporary mesopotamians to assess the relative values of their other commodities. Babylonians had 1 gur of barley ideally priced at 1 shekel even when the measure of gur changed. The price still fluctuated and the equivalency was not an instance of governmental price control. But it is an example of a bimonetary currency-mechanism and a curious form of silver standard.

Physically owning silver was not necessary for non-barter transactions. The imported metal available to the Mesopotamian economy probably would have been far too scarce in any case. Internal trade relied on credit and utilized unit equivalencies. This bimonetary system enabled credit relationships that could be expressed in both precious metal and grain. Loans concerning goods or services could stipulate that the anticipated silver was to be repaid with barley. There was silver in circulation but as a commodity easily stored long-term, it served better for things such as paying taxes and rents to palaces. Some palaces used entrepreneurial middlemen who collected produce as payment and paid the palace in silver.

Now remember the realities of premodern agriculture. It wasn't always possible for everyone to pay fully in time. To alleviate this (and with interest, to profit on), middlemen could extend credit to payers and palaces likewise to middlemen. A buffered system of trust backed with partly circulating, partly stored, inherently valued precious metal that also served as a valuational reference point for other commodities. Then add grain to the currency-system so that value-equivalent physical transfers are possible at all societal levels, anywhere and anytime.

It's quite easy to imagine all this working as an internal system, but how about inter-state trade? Credit and uncoined precious metals again. An Old Assyrian private archive from around 20th century BC documents Assyrian traders at Kanesh (Central Anatolia) importing Babylonian textiles and eastern tin, being paid with silver and gold, and then shipping the metal to Assur (Tigris river valley). These traders belonged to a state-dictated cooperative enterprise spanning multiple Anatolian cities, kārum, that operated on credit and book transfers between members. Accounts were then periodically settled at the enterprise AND palatial-taxation level.

Both the bimonetary (silver and barley) currency-system and credit were crucial at arranging the internal palatial economy and financing external trade. Coinage was not strictly needed and it's not obvious it would have added all that much to the system. You would see more of the same in coin-scarce early modern Europe, from informal credit between commoners, to ledger-money, to state debt -expansed money supply. All this of course would require tight coordination and robust capability for record-making and -keeping. At least some Mesopotamian regimes truly excelled at this, attested by the fact that there are well over 100k administrative cuneiform tablets preserved from the sixty year period of Ur III alone.

In case of follow-up questions springing up, I have very little time to spare at the moment, so probably can't answer (in depth and for a while at least).

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Sources:

Hudson, 2020, Origins of Money and Interest: Palatial Credit, Not Barter (in Battilossi, Cassis & Yago, eds., Handbook of the History of Money and Currency)

Jursa & García, 2015, The ancient Near East and Egypt (in Monson & Scheidel, eds., Fiscal Regimes and the Political Economy of Premodern States)

Van De Mieroop, 2014, Silver as a Financial Tool in Ancient Egypt and Mesopotamia (in Bernholz & Vaubel, eds., Explaining Monetary and Financial Innovation: A Historical Analysis)

Veenhof, 2011, The Archives of Old Assyrian Traders: their Nature, Functions and Use (in Faraguna, ed., Archives and archival documents in ancient societies: Legal documents in ancient societies IV)